Help with transitioning color with dynamic paint
First Suggestions.



Select for Brushes, for data value Paint Source

Mesh Volume + Proximity 

as indicated by the Yellow Arrow. Image Above.



For the Canvas change Resolution to 128 for a higher resolution image and smoother transtions

See Yellow Arrow Image above.

I am not claiming this is the minimal set of changes.  You can test the two changes.

This is an improvement.  There may be other lesser problems.





OLDER


To have a circular or blob spread,  create a sphere or similar blob and use it as your paint brush. Your current box paintbrush is not going to fulfill your goals.  A sphere is a standard object and can be further shaped with proportional edit tool which can be searched here at BSE or the general internet.  This low cost improvement can be started by you immediately.
Experiment with the Dry and Dissolve features of Dynamic Paint.


Other suggestions may come later.  My time currently at BSE is expiring.